\(\int \frac {(a+c x^2)^p}{d+e x+f x^2} \, dx\) [178]

Optimal result

Integrand size = 22, antiderivative size = 412 \[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=-\frac {2 f x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {c x^2}{a},\frac {4 f^2 x^2}{\left (e-\sqrt {e^2-4 d f}\right )^2}\right )}{e^2-4 d f-e \sqrt {e^2-4 d f}}-\frac {2 f x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {c x^2}{a},\frac {4 f^2 x^2}{\left (e+\sqrt {e^2-4 d f}\right )^2}\right )}{e^2-4 d f+e \sqrt {e^2-4 d f}}-\frac {2 f^2 \left (a+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {4 f^2 \left (a+c x^2\right )}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2}\right )}{\sqrt {e^2-4 d f} \left (4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2\right ) (1+p)}+\frac {2 f^2 \left (a+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {4 f^2 \left (a+c x^2\right )}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2}\right )}{\sqrt {e^2-4 d f} \left (4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2\right ) (1+p)} \] Output:

-2*f*x*(c*x^2+a)^p*AppellF1(1/2,-p,1,3/2,-c*x^2/a,4*f^2*x^2/(e-(-4*d*f+e^2 
)^(1/2))^2)/(e^2-4*d*f-e*(-4*d*f+e^2)^(1/2))/((1+c*x^2/a)^p)-2*f*x*(c*x^2+ 
a)^p*AppellF1(1/2,-p,1,3/2,-c*x^2/a,4*f^2*x^2/(e+(-4*d*f+e^2)^(1/2))^2)/(e 
^2-4*d*f+e*(-4*d*f+e^2)^(1/2))/((1+c*x^2/a)^p)-2*f^2*(c*x^2+a)^(p+1)*hyper 
geom([1, p+1],[2+p],4*f^2*(c*x^2+a)/(4*a*f^2+c*(e-(-4*d*f+e^2)^(1/2))^2))/ 
(-4*d*f+e^2)^(1/2)/(4*a*f^2+c*(e-(-4*d*f+e^2)^(1/2))^2)/(p+1)+2*f^2*(c*x^2 
+a)^(p+1)*hypergeom([1, p+1],[2+p],4*f^2*(c*x^2+a)/(4*a*f^2+c*(e+(-4*d*f+e 
^2)^(1/2))^2))/(-4*d*f+e^2)^(1/2)/(4*a*f^2+c*(e+(-4*d*f+e^2)^(1/2))^2)/(p+ 
1)
 

Mathematica [F]

\[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=\int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx \] Input:

Integrate[(a + c*x^2)^p/(d + e*x + f*x^2),x]
 

Output:

Integrate[(a + c*x^2)^p/(d + e*x + f*x^2), x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx\)

\(\Big \downarrow \) 1326

\(\displaystyle \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2}dx\)

Input:

Int[(a + c*x^2)^p/(d + e*x + f*x^2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1326

Int[((a_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_ 
Symbol] :> Unintegrable[(a + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, 
c, d, e, f, p, q}, x] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]
 

Maple [F]

\[\int \frac {\left (c \,x^{2}+a \right )^{p}}{f \,x^{2}+e x +d}d x\]

Input:

int((c*x^2+a)^p/(f*x^2+e*x+d),x)
 

Output:

int((c*x^2+a)^p/(f*x^2+e*x+d),x)
 

Fricas [F]

\[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{f x^{2} + e x + d} \,d x } \] Input:

integrate((c*x^2+a)^p/(f*x^2+e*x+d),x, algorithm="fricas")
 

Output:

integral((c*x^2 + a)^p/(f*x^2 + e*x + d), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=\text {Timed out} \] Input:

integrate((c*x**2+a)**p/(f*x**2+e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{f x^{2} + e x + d} \,d x } \] Input:

integrate((c*x^2+a)^p/(f*x^2+e*x+d),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + a)^p/(f*x^2 + e*x + d), x)
 

Giac [F]

\[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{f x^{2} + e x + d} \,d x } \] Input:

integrate((c*x^2+a)^p/(f*x^2+e*x+d),x, algorithm="giac")
 

Output:

integrate((c*x^2 + a)^p/(f*x^2 + e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=\int \frac {{\left (c\,x^2+a\right )}^p}{f\,x^2+e\,x+d} \,d x \] Input:

int((a + c*x^2)^p/(d + e*x + f*x^2),x)
 

Output:

int((a + c*x^2)^p/(d + e*x + f*x^2), x)
 

Reduce [F]

\[ \int \frac {\left (a+c x^2\right )^p}{d+e x+f x^2} \, dx=\int \frac {\left (c \,x^{2}+a \right )^{p}}{f \,x^{2}+e x +d}d x \] Input:

int((c*x^2+a)^p/(f*x^2+e*x+d),x)
 

Output:

int((a + c*x**2)**p/(d + e*x + f*x**2),x)